A proposal for realising quantum scars in the tilted 1D Fermi-Hubbard model
Jean-Yves Desaules, Ana Hudomal, Christopher J. Turner, Zlatko Papi?
AA proposal for realising quantum scars in the tilted 1D Fermi-Hubbard model
Jean-Yves Desaules, Ana Hudomal,
1, 2
Christopher J. Turner, and Zlatko Papi´c School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom Institute of Physics Belgrade, University of Belgrade, 11080 Belgrade, Serbia (Dated: February 3, 2021)Motivated by recent observations of ergodicity breaking due to Hilbert space fragmentation in1D Fermi-Hubbard chains with a tilted potential [Scherg et al. , arXiv:2010.12965], we show thatthe same system also hosts quantum many-body scars in a regime U ≈ ∆ (cid:29) J at electronic fillingfactor ν = 1. We numerically demonstrate that the scarring phenomenology in this model is similarto other known realisations such as Rydberg atom chains, including persistent dynamical revivalsand ergodicity-breaking many-body eigenstates. At the same time, we show that the mechanism ofscarring in the Fermi-Hubbard model is different from other examples in the literature: the scarsoriginate from a subgraph, representing a free spin-1 paramagnet, which is weakly connected to therest of the Hamiltonian’s adjacency graph. Our work demonstrates that correlated fermions in tiltedoptical lattices provide a platform for understanding the interplay of many-body scarring and otherforms of ergodicity breaking, such as localisation and Hilbert space fragmentation. Introduction.
Recently, there has been much interestin understanding how closed many-body quantum sys-tems evolve in time when taken out of their equilibriumstate. While many such systems rapidly return to theirequilibrium state, in accordance with fundamental princi-ples of quantum statistical mechanics [1], much of recentwork has focused on systems that fail to do so as a con-sequence of ergodicity breaking [2, 3], either due to thespecial mathematical structure known as integrability orvery strong disorder which leads to (many-body) locali-sation. Both of these paradigms of behaviour are activelyinvestigated in experiments on cold atoms, trapped ionsand superconducting qubits [4–7].The inability of non-ergodic systems to act as heatreservoirs for their smaller parts has been traditionallyknown to affect the entire spectrum of the system. Re-cently, however, there has been a flurry of interest in weak ergodicity breaking phenomena [8]. The latter refersto the emergence of a dynamically-decoupled subspacewithin the many-body Hilbert space, in general with-out any underlying symmetry, spanned by ergodicity-breaking eigenstates. This behaviour was first theo-retically established in the Affleck-Kennedy-Lieb-Tasaki(AKLT) model [9, 10], followed by the discovery of sim-ilar phenomenology in other non-integrable lattice mod-els [11–18], models of correlated fermions and bosons [19–25], frustrated magnets [26, 27], topological phases ofmatter [28, 29], and periodically driven systems [30–34].In these examples, the ergodicity-breaking eigenstatesare either explicitly embedded into a many-body spec-trum via the mechanism due to Shiraishi and Mori [35],or they form a representation of an algebra [36–38].In recent experiments on Rydberg atom arrays [39],weak ergodicity breaking was observed via persistent re-vivals following the global quench of the system. Thisdynamics, which has also been interpreted as preces-sion of an emergent su(2) spin [40, 41], bears the anal-ogy with the dynamics of a semiclassical wavepacket in- side a chaotic stadium billiard [42], motivating the name“quantum many-body scarring” [43–45]. More recently,quantum many-body scarring has been shown to occur inhigher dimensions [46–48] and in the presence of certainkinds of perturbations [49–51] including disorder [52].On the other hand, it has also been shown that ergod-icity breaking can occur due to a fracturing of the Hilbertspace into dynamically disconnected components [53–56].This typically occurs by the interplay of a locality re-striction, i.e. terms with support strictly only on smallintervals of sites, together with a higher-moment sym-metry such as charge dipole conservation, which non-trivially intertwines spatial and internal symmetries. Re-cent work [57] has demonstrated that Hilbert space frag-mentation can be experimentally realised via magneticfield gradient applied to the Fermi-Hubbard model in a1D optical lattice. Apart from offering a new platformto investigate the link between fragmentation and the so-called Stark many-body localisation [58–60], an immedi-ate question presents itself: can the tilted Fermi-Hubbardmodel realise quantum many-body scars?In this paper, we show that quantum many-body scarsarise in the limit U ≈ ∆ (cid:29) J in the tilted Fermi-Hubbardmodel, and that they can be detected using the quenchfrom a specific initial state at a different filling factorfrom the one considered in Ref. 57. We derive an effectivemodel for this setup, which can be mapped to a spinfulgeneralisation of the fractional quantum Hall effect on athin torus [23], allowing for a practical experimental re-alisation. While the phenomenology of quantum many-body scars is shown to be largely similar to their realisa-tion in Rydberg atom systems [39], including in particu-lar an extensive set of eigenstates which violate the Eigen-state Thermalisation Hypothesis (ETH) [61, 62], the ori-gin of scars is different in the two systems and can beintuitively understood from a graph-theoretic viewpoint. Large-tilt limit of the Fermi-Hubbard model.
The 1D a r X i v : . [ c ond - m a t . s t r- e l ] F e b Fermi-Hubbard model is given by the Hamiltonianˆ H = (cid:88) j, σ = ↑ , ↓ (cid:16) − J ˆ c † j,σ ˆ c j +1 ,σ + h . c . + ∆ j ˆ n j,σ (cid:17) + U (cid:88) j ˆ n j, ↑ ˆ n j, ↓ , (1)where ˆ c † j,σ denotes the usual electron creation operator onsite j with spin projection σ , ˆ n j,σ ≡ ˆ c † j,σ ˆ c j,σ , J and U arethe hopping and on-site interaction terms, respectively.Tilt of the optical lattice is parametrised by ∆, which wetake to be spin-independent [57]. Note that tilting hasthe structure of a dipole term, ∼ j ˆ n j . Below we imposeopen boundary conditions on the model in Eq. (1), andrestrict to the electron filling factor ν = 1, i.e., with N/ ↑ and N/ ↓ on achain of N sites (assumed to be even). We also set J = 1for simplicity. We label the Fock states using ↑ to denotea fermion with spin up and ↓ with spin down, while 0stands for an empty site and (cid:108) denotes a doublon.We focus on the regime ∆ ≈ U (cid:29) J . In this case thesum of the dipole moment and the number of doublons iseffectively conserved. The dominant contribution to theHamiltonian (using a Schrieffer-Wolff transformation atfirst order [63]) is then given byˆ H eff = − J (cid:88) j,σ ˆ c † j,σ ˆ c j +1 ,σ ˆ n j,σ (1 − ˆ n j +1 ,σ ) + h . c . + ( U − ∆) (cid:88) j ˆ n j, ↑ ˆ n j, ↓ . (2)In this effective Hamiltonian, hopping to the left (whichdecreases the total dipole moment by 1) is only allowed ifit increases the number of doublons by the same amount(¯ σ denotes opossite spin from σ ).The action of the Hamiltonian (2) within the ν = 1 sec-tor fragments the Hilbert space beyond the simple con-servation of U + ∆. In this work we focus on the largestconnected component, which is the one containing thestate with alternating ↑ and ↓ fermions. In addition tothe symmetries of the full model in Eq. (1), i.e., SU(2)spin symmetry and spin reversal [64], the Hamiltonian(2) projected to the largest sector has an additional sym-metry related to spatial inversion and particle-hole ex-change [65]. After resolving these symmetries, we findthe level statistics parameter (cid:104) r (cid:105) [66] to be close to 0.53for all symmetry sectors with large numbers of states( (cid:38) ) [65]. From these values which coincide withthe Wigner-Dyson statistics [67], we expect the modelin Eq. (2) to be chaotic. We next outline an intuitiveapproach for identifying many-body scars in this model. Embedded hypergrid subgraph.
A practical diagnosticof quantum many-body scars is the existence of weakly-correlated states which undergo robust revivals underquench dynamics, while the majority of other initialstates thermalise fast and do not display revivals. In | − + i , | + −i| ↓ ↑i , | ↑ ↓i HypergridOther states
Figure 1. Adjacency graph of the effective model in Eq. (2)for N = 6. Red vertices denote the states belonging to the hy-pergrid, with the black vertices corresponding to |− + (cid:105) , | + −(cid:105) states defined in the text. Green vertices are the isolatedstates | ↓ o ↑(cid:105) , | ↑ o ↓(cid:105) which live on the tails of the graph.For this graph, hypergrid contains 27 vertices out of 63. scarred Rydberg atom chains [39], the reviving statescan be identified by viewing the system as a constrainedspin-1 / . . . ↑↑ . . . are energetically prohibited [68]. The revivingN´eel state, ↑↓↑↓ . . . , is the densest configuration compat-ible with the constraint, and it is an extremal vertex ofthe Hamiltonian adjacency graph [43]. We next show, byexamining the adjacency graph of the model in Eq. (2),that we can identify a subgraph, weakly coupled to therest of the Hilbert space, which contains the reviving ini-tial states and leaves a strong imprint on the scarredeigenstates. This leads to a transparent manifestation ofscarring in the original Fock basis, in contrast with Ry-dberg atoms where scarred dynamics leads to spreadingacross the entire adjacency graph [49].In Fig. 1 we plot the adjacency graph of the Hamilto-nian in Eq. (2) for a small system. Each vertex corre-sponds to a product state, and two states are connectedby an edge if the matrix element between them is non-zero. For the effective model in Eq. (2), it is possible togauge away the fermionic minus signs [65], resulting in anunweighted, undirected graph. As the Hamiltonian (2)(for U = ∆) has no diagonal elements and the spectrum issymmetric around zero, all product states are effectivelyin the infinite temperature ensemble and are expectedto thermalise quickly. As we confirm numerically below,there are two important exceptions.First, as highlighted in red colour in Fig. 1, there isa regular subgraph which has the form of the hyper-grid – a Cartesian product of line graphs (in our case,of length 3), i.e., the hypergrid is isomorphic to an adja-cency graph of a free spin-1 paramagnet. This mappingcan be understood by looking at the state | ↓↑↓↑↓↑ . . . (cid:105) .Each cell of two sites can take the values − := ↓↑ , o := (cid:108) ↑↓ , leading to a three level system. Note thatthe configuration 0 (cid:108) is omitted, as doublons can onlybe formed by hopping to the left. On the other hand,hopping between two neighbouring cell will break thismapping and take the system out of the hypergrid sub-graph. Inside the hypergrid, we identify two states forwhich the cell alternates between − and +. These are thestate |− + (cid:105) := |− + − + . . . (cid:105) = | ↓↑↑↓↓↑↑↓ . . . (cid:105) and its spin-inverted partner, | + −(cid:105) = | + − + − . . . (cid:105) := | ↑↓↓↑↑↓↓↑ . . . (cid:105) .The states |− + (cid:105) and | + −(cid:105) for N = 6 are shown in blackcolour in Fig. 1. These two states are the only cornersof the hypergrid (state with only + and − cells) withno edges going out of it. As we show below, either ofthese states shows persistent oscillations in quench dy-namics, undergoing robust state transfer to their spin-inverted counterpart. While other corners of the hyper-grid also show revivals, they are much smaller in am-plitude and decay faster due to the leakage out of thissubstructure. The second example of a reviving stateis | ↓ o ↑(cid:105) := | ↓↓ . . . ↓(cid:108) ↑↑ . . . ↑(cid:105) (and its spin-reversedpartner | ↑ o ↓(cid:105) ), which is situated on a tail-like struc-ture of length 3 (independent of system size) with mini-mal connectivity to the rest of the Hilbert space (greenpoints in Fig 1). Similar tail-like structures occur inconstrained spin models [69]. Many-body scarred dynamics and eigenstates.
Havingidentified candidate states for revivals, we now scrutinisetheir quench dynamics using large-scale exact diagonali-sation simulations of the effective model in Eq. (2). Mak-ing use of various symmetries present in the model, wehave been able to exactly simulate dynamics for up to N = 22 electrons. For convenience, the simulations wereperformed in the spin representation of the model [65].Fig. 2(a) shows the time dependence of the entangle-ment entropy S ent ( t ) when the system is quenched fromvarious initial product states, such as | − + (cid:105) , | ↓ o ↑(cid:105) anda few randomly-chosen product states. S ent is definedas the von Neumann entropy of the reduced density ma-trix for one half of the chain. In all cases, entropy growslinearly in time, consistent with thermalisation of the sys-tem. However, the coefficient of linear growth is visiblydifferent for | − + (cid:105) and | ↓ ↑(cid:105) states, and it is smallerthan that of random states, indicating non-ergodic dy-namics. The hallmark of many-body scars are the oscil-lations superposed on top of the linear growth, as seenin the scarred dynamics in Rydberg atom chains [43].Rapid growth of entropy at short times is a direct resultof the structure of initial states and the location of thebipartition. When starting in | ↓ o ↑(cid:105) , the only possiblemoves create entanglement between the two sites closestto the bipartition on either side. Similarly, for |− + (cid:105) statewith N = 18 the bipartition cuts through the cell form-ing an effective spin-1. Shifting the cut by one site oneither side moves it between two cells and leads to a slowgrowth of entropy at short time for this state.Entropy oscillations mirror those of the wavefunctionreturn probability, |(cid:104) ψ | e − i ˆ Ht | ψ (cid:105)| , plotted in Fig. 2(b).For the isolated state | ↓ o ↑(cid:105) , only a single wavefunc-tion revival is clearly visible as the return probabilitydecays rapidly once the wavefunction leaks out of the S e n t (a) |− + i |↓ o ↑i Random states . . . | h ψ | e − i ˆ H t | ψ i | (b) |h + −| e − i ˆ Ht |− + i| t . . . P H G ( t ) (c) . . /N . . F i d . d e n s . |− + i / D Figure 2. Dynamics in the effective model (2) for N = 18 for |− + (cid:105) , | ↓ o ↑(cid:105) and randomly chosen initial states. (a) Entan-glement entropy S ent for an equal bipartition of the system.Entropy grows linearly in time for all states, consistent withthermalising dynamics, but it shows oscillations due to many-body scarring. The black line correspond to the average over10 random product states and the shaded area is the standarddeviation. (b) Fidelity dynamics for the same initial states asin (a). Inset shows the finite-size scaling of the fidelity den-sity at the first revival for |− + (cid:105) state, demonstrating a muchhigher value than expected for a random state. Blue dottedline shows the amplitude of state transfer between |− + (cid:105) and | + −(cid:105) states. (c) Probability to remain within the hypergridover time is much higher for |− + (cid:105) than other states. tail of the graph. Because of the low connectivity ofthe tail, the first revival is still visible on the scale ofFig. 2(b). The revival time can be accurately estimatedby assuming the tail is completely disconnected, lead-ing to the period π/ √
2. In contrast, the state |− + (cid:105) displays several revivals with the sizable weight of thewavefunction ∼
40% returning to its initial value. Thefidelity density, − N ln |(cid:104) ψ | e − i ˆ Ht | ψ (cid:105)| , shown in the in-set, converges as 1 /N to a value of 0 . D − expected for arandom state leads to a fidelity density of 0 . |− + (cid:105) and its partner | + −(cid:105) ,illustrated by the dotted line in Fig. 2(b). From the hy-pergrid analysis, we expect the revival period to be √ π ,coming from the 2 π period of free precession and the factthat the spin-1 operators have an energy scale √
2. Thisprediction closely matches the revival period observed inFig. 2(b).The importance of the hypergrid subgraph for thescarred dynamics is illustrated in Fig. 2(c) which plotsthe probability to remain in the hypergrid, P HG ( t ) = (cid:104) ψ | e i ˆ Ht ˆ P HG e − i ˆ Ht | ψ (cid:105) , where ˆ P HG is the projector ontothe subspace spanned by product states belonging to thehypergrid. For the initial state |− + (cid:105) , we observe that thewavefunction remains concentrated inside the hypergrid,even at late times. This is in stark contrast with thePXP model [49], where the wavefunction spreads acrossthe entire graph by the time it undergoes the first revival.Furthermore, even at the first revival peak the fidelity islower than the value of P HG . This shows that the wave-function does not exactly return to itself but gets morespread even within the hypergrid. Finally, for this ini-tial state after a long time P HG converges to a non-zerovalue which is higher than expected from the relative sizeof the hypergrid in the Hilbert space [65], hinting that thegraph could have some additional structure that preventsstates from leaking out of the hypergrid. Figure 3. Eigenstate properties of the effective model (2).(a) Overlap of eigenstates with the |− + (cid:105) state as a function oftheir energy E . (b) Entanglement entropy S ent of the eigen-states. Data is for system size N = 16. Red dots correspondto eigenstates with total spin S = 1, while the blue ones onesmark all other spin values. The squares indicate the eigen-states sitting at the top of each tower of states. These towershave an energy separation of approximately √
2, as expectedfor the spin-1 hypergrid.
Properties of eigenstates of the model (2) are sum-marised in Fig. 3. The projection of eigenstates onto |− + (cid:105) state, shown in panel (a), displays distinct towerstructures reminiscent of other scarred models [13, 43].The existence of towers implies that eigenstates tendto concentrate around certain energies in the spectrum,causing an ETH violation. The separation between thetowers is approximately ∆ E ≈ √
2, as expected fromthe embedded hypergrid. Note that the eigenstates havebeen classified according to the conserved total value ofspin S ; in contrast, |− + (cid:105) state is not an eigenstate of S .One can show that for this state, (cid:104) S (cid:105) = N/
2, thus |− + (cid:105) is predominantly supported by S = 1 and S = 2 eigen-states at the given system size. The S = 1 eigenstates are indicated by red points in Fig. 3.Similar violation of the ETH can be seen in the largespread in entanglement entropy of eigenstates shown inFig. 3(b), showing that eigenstates of similar energy havevery different amounts of entanglement. Part of thisspreading, however, can be attributed to the eigenstatesbelonging to different spin sectors S , giving rise to multi-ple bands that do not fully overlap at the system sizeshown in Fig. 3(b) [65]. The distribution of entropyin the S = 1 sector [red points in Fig. 3(b)] is rela-tively narrow apart from two “outliers” shown at energy E ≈ ±√
2, which sit at the top of the tower for theirsector in Fig. 3(a). The states at the top of each towerare indicated by squares, but unlike the PXP model [49]these states are less clearly separated in overlap with theother states in the same tower.
Experimental implications.
The effective model stud-ied above is exact for U = ∆ → ∞ . For prospective ex-perimental realisations, it is important to ascertain thatthe same physics persists for accessible values of U , ∆and that it can be detected using local measurements,as measuring entropy or quantum fidelity is more chal-lenging. We demonstrate this in Fig. 4 for the full modelin Eq. (1) focusing on the regime U, ∆ <
10. Panel (a)shows the dynamics of imbalance on the even/odd sub-lattices, I = ( N o − N e ) / ( N o + N e ), where N e / o is thetotal number of fermions on the even or odd sites. Theimbalance is bounded between -1 and 1. For simplicity,here we consider system sizes divisible by 4, for whichthe imbalance for the initial state |− + (cid:105) is zero. We seerobust oscillations in I with the frequency that matcheshalf the wavefunction revival frequency in Fig. 2(b). Theamplitude of the imbalance revival remains close to theinfinite-limit value for U = ∆ (cid:38)
6. In order to furtherconfirm that the hypergrid is causing non-ergodicity inthe full model we devised a simple perturbation based onthe Hilbert space structure of the effective model. Ap-plying this perturbation to the full tilted Fermi-Hubbardchain leads to an improvement of revivals, consistent with U = ∆ = ∞ limit [65]. t . . . . I (a) ∆=3∆=4 . ∞ U ∆ (b) . . . I Figure 4. (a) Occupation imbalance for various values of U = ∆ for the full model in Eq. (1) for system size N =12.(b) U − ∆ phase diagram showing the scarring regime nearthe diagonal (dashed line). The colour scale represents thevalue of the first peak of the imbalance for N =12. Conclusions and discussion.
In this paper we have pre-sented a proposal for experimentally realising quantummany-body scars in the regime U = ∆ of the tilted Fermi-Hubbard model. We have identified product states |− + (cid:105) , | + −(cid:105) at filling factor ν = 1 which give rise to scarred dy-namics and reveal towers of ergodicity-breaking many-body eigenstates in the spectra of such systems, allowingto investigate the interplay of many-body scarring withother facets of weak ergodicity breaking such as locali-sation and Hilbert space fragmentation. In addition tothe filling factor ν = 1, we have also studied the fill-ing ν = 1 / (cid:29) U, J and going to the third or-der of Schrieffer-Wolff, we found analogous signatures ofscars [65], provided we neglect the diagonal terms in theeffective Hamiltonian. Under these assumptions, the re-sulting model can be viewed as a spinful generalisationof the fractional quantum Hall effect on a thin torus [23].By contrast, the approach presented here for ν = 1 isconsiderably simpler as it allows to conveniently elimi-nate the undesirable diagonal terms. Acknowledgements.
We ackowledge support by theLeverhulme Trust Research Leadership Award RL-2019-015 (ZP, AH), and by EPSRC grants EP/R020612/1(ZP), EP/R513258/1 (JYD), and EP/M50807X/1(CJT). Statement of compliance with EPSRC policyframework on research data: This publication is theoret-ical work that does not require supporting research data.A.H. acknowledges funding provided by the Institute ofPhysics Belgrade, through the grant by the Ministry ofEducation, Science, and Technological Development ofthe Republic of Serbia. Part of the numerical simula-tions were performed on the PARADOX-IV supercom-puting facility at the Scientific Computing Laboratory,National Center of Excellence for the Study of ComplexSystems, Institute of Physics Belgrade. [1] Christian Gogolin and Jens Eisert, “Equilibration, ther-malisation, and the emergence of statistical mechanics inclosed quantum systems,” Rep. Prog. Phys. , 056001(2016).[2] Rahul Nandkishore and David A Huse, “Many-body lo-calization and thermalization in quantum statistical me-chanics,” Annu. Rev. Condens. Matter Phys. , 15–38(2015).[3] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, andMaksym Serbyn, “Colloquium: Many-body localization,thermalization, and entanglement,” Rev. Mod. Phys. ,021001 (2019).[4] Toshiya Kinoshita, Trevor Wenger, and David S. Weiss,“A quantum Newton’s cradle,” Nature , 900 (2006).[5] Michael Schreiber, Sean S. Hodgman, Pranjal Bordia,Henrik P. L¨uschen, Mark H. Fischer, Ronen Vosk, EhudAltman, Ulrich Schneider, and Immanuel Bloch, “Obser-vation of many-body localization of interacting fermions in a quasirandom optical lattice,” Science , 842(2015).[6] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W.Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Mon-roe, “Many-body localization in a quantum simulatorwith programmable random disorder,” Nat. Phys. ,907 (2016).[7] B. Chiaro, C. Neill, A. Bohrdt, M. Filippone, F. Arute,K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends,S. Boixo, D. Buell, B. Burkett, Y. Chen, Z. Chen,R. Collins, A. Dunsworth, E. Farhi, A. Fowler, B. Foxen,C. Gidney, M. Giustina, M. Harrigan, T. Huang,S. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi,J. Kelly, P. Klimov, A. Korotkov, F. Kostritsa, D. Land-huis, E. Lucero, J. McClean, X. Mi, A. Megrant,M. Mohseni, J. Mutus, M. McEwen, O. Naaman, M. Nee-ley, M. Niu, A. Petukhov, C. Quintana, N. Rubin,D. Sank, K. Satzinger, A. Vainsencher, T. White,Z. Yao, P. Yeh, A. Zalcman, V. Smelyanskiy, H. Neven,S. Gopalakrishnan, D. Abanin, M. Knap, J. Martinis,and P. Roushan, “Direct measurement of non-local in-teractions in the many-body localized phase,” (2020),arXiv:1910.06024 [cond-mat.dis-nn].[8] Maksym Serbyn, Dmitry A. Abanin, and Zlatko Papi´c,“Quantum many-body scars and weak breaking of ergod-icity,” (2020), arXiv:2011.09486 [quant-ph].[9] Daniel P Arovas, “Two exact excited states for the s = 1AKLT chain,” Physics Letters A , 431–433 (1989).[10] Sanjay Moudgalya, Nicolas Regnault, and B. An-drei Bernevig, “Entanglement of exact excited statesof Affleck-Kennedy-Lieb-Tasaki models: Exact results,many-body scars, and violation of the strong eigenstatethermalization hypothesis,” Phys. Rev. B , 235156(2018).[11] Michael Schecter and Thomas Iadecola, “Weak ergodic-ity breaking and quantum many-body scars in spin-1 xy magnets,” Phys. Rev. Lett. , 147201 (2019).[12] Thomas Iadecola and Michael Schecter, “Quantummany-body scar states with emergent kinetic constraintsand finite-entanglement revivals,” Phys. Rev. B ,024306 (2020).[13] Kieran Bull, Ivar Martin, and Z. Papi´c, “Systematicconstruction of scarred many-body dynamics in 1d latticemodels,” Phys. Rev. Lett. , 030601 (2019).[14] Sambuddha Chattopadhyay, Hannes Pichler, Mikhail D.Lukin, and Wen Wei Ho, “Quantum many-body scarsfrom virtual entangled pairs,” Phys. Rev. B , 174308(2020).[15] Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Kat-sura, “Onsager’s scars in disordered spin chains,” Phys.Rev. Lett. , 180604 (2020).[16] Sanjay Moudgalya, Edward O’Brien, B. Andrei Bernevig,Paul Fendley, and Nicolas Regnault, “Large classesof quantum scarred Hamiltonians from matrix productstates,” Phys. Rev. B , 085120 (2020).[17] Federica Maria Surace, Giuliano Giudici, and MarcelloDalmonte, “Weak-ergodicity-breaking via lattice super-symmetry,” Quantum , 339 (2020).[18] Yoshihito Kuno, Tomonari Mizoguchi, and YasuhiroHatsugai, “Flat band quantum scar,” Phys. Rev. B ,241115 (2020).[19] Oskar Vafek, Nicolas Regnault, and B. Andrei Bernevig,“Entanglement of Exact Excited Eigenstates of the Hub-bard Model in Arbitrary Dimension,” SciPost Phys. ,
043 (2017).[20] Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich,“Unified structure for exact towers of scar states inthe affleck-kennedy-lieb-tasaki and other models,” Phys.Rev. B , 195131 (2020).[21] Daniel K. Mark and Olexei I. Motrunich, “ η -pairingstates as true scars in an extended Hubbard model,”Phys. Rev. B , 075132 (2020).[22] Sanjay Moudgalya, Nicolas Regnault, and B. AndreiBernevig, “ η -pairing in Hubbard models: From spectrumgenerating algebras to quantum many-body scars,” Phys.Rev. B , 085140 (2020).[23] Sanjay Moudgalya, B. Andrei Bernevig, and NicolasRegnault, “Quantum many-body scars in a landau levelon a thin torus,” Phys. Rev. B , 195150 (2020).[24] Ana Hudomal, Ivana Vasi´c, Nicolas Regnault, andZlatko Papi´c, “Quantum scars of bosons with correlatedhopping,” Commun. Phys. , 99 (2020).[25] Hongzheng Zhao, Joseph Vovrosh, Florian Mintert, andJohannes Knolle, “Quantum many-body scars in opticallattices,” Phys. Rev. Lett. , 160604 (2020).[26] Kyungmin Lee, Ronald Melendrez, Arijeet Pal, andHitesh J. Changlani, “Exact three-colored quantum scarsfrom geometric frustration,” Phys. Rev. B , 241111(2020).[27] Paul A. McClarty, Masudul Haque, Arnab Sen, andJohannes Richter, “Disorder-free localization and many-body quantum scars from magnetic frustration,” Phys.Rev. B , 224303 (2020).[28] Seulgi Ok, Kenny Choo, Christopher Mudry, Clau-dio Castelnovo, Claudio Chamon, and Titus Neupert,“Topological many-body scar states in dimensions one,two, and three,” Phys. Rev. Research , 033144 (2019).[29] Julia Wildeboer, Alexander Seidel, N. S. Srivatsa, AnneE. B. Nielsen, and Onur Erten, “Topological quan-tum many-body scars in quantum dimer models onthe kagome lattice,” (2020), arXiv:2009.00022 [cond-mat.str-el].[30] Berislav Buca, Joseph Tindall, and Dieter Jaksch,“Non-stationary coherent quantum many-body dynamicsthrough dissipation,” Nat. Commun. , 1730 (2019).[31] J. Tindall, B. Buˇca, J. R. Coulthard, and D. Jaksch,“Heating-induced long-range η pairing in the Hubbardmodel,” Phys. Rev. Lett. , 030603 (2019).[32] Sho Sugiura, Tomotaka Kuwahara, and Keiji Saito,“Many-body scar state intrinsic to periodically drivensystem: Rigorous results,” (2019), arXiv:1911.06092[cond-mat.stat-mech].[33] Kaoru Mizuta, Kazuaki Takasan, and Norio Kawakami,“Exact Floquet quantum many-body scars under Ryd-berg blockade,” Phys. Rev. Research , 033284 (2020).[34] Bhaskar Mukherjee, Sourav Nandy, Arnab Sen, Dip-timan Sen, and K. Sengupta, “Collapse and revivalof quantum many-body scars via Floquet engineering,”Phys. Rev. B , 245107 (2020).[35] Naoto Shiraishi and Takashi Mori, “Systematic construc-tion of counterexamples to the eigenstate thermalizationhypothesis,” Phys. Rev. Lett. , 030601 (2017).[36] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R.Klebanov, “Many-body scars as a group invariant sectorof hilbert space,” Phys. Rev. Lett. , 230602 (2020).[37] Jie Ren, Chenguang Liang, and Chen Fang, “Quasi-symmetry groups and many-body scar dynamics,”(2020), arXiv:2007.10380 [cond-mat.str-el]. [38] Nicholas O’Dea, Fiona Burnell, Anushya Chandran, andVedika Khemani, “From tunnels to towers: Quantumscars from lie algebras and q -deformed lie algebras,”Phys. Rev. Research , 043305 (2020).[39] Hannes Bernien, Sylvain Schwartz, Alexander Keesling,Harry Levine, Ahmed Omran, Hannes Pichler, Soon-won Choi, Alexander S. Zibrov, Manuel Endres, MarkusGreiner, Vladan Vuletic, and Mikhail D. Lukin, “Prob-ing many-body dynamics on a 51-atom quantum simula-tor,” Nature , 579 (2017).[40] Soonwon Choi, Christopher J. Turner, Hannes Pich-ler, Wen Wei Ho, Alexios A. Michailidis, Zlatko Papi´c,Maksym Serbyn, Mikhail D. Lukin, and Dmitry A.Abanin, “Emergent su(2) dynamics and perfect quantummany-body scars,” Phys. Rev. Lett. , 220603 (2019).[41] Kieran Bull, Jean-Yves Desaules, and Zlatko Papi´c,“Quantum scars as embeddings of weakly broken lie alge-bra representations,” Phys. Rev. B , 165139 (2020).[42] Eric J. Heller, “Bound-state eigenfunctions of classicallychaotic hamiltonian systems: Scars of periodic orbits,”Phys. Rev. Lett. , 1515 (1984).[43] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,and Z. Papic, “Weak ergodicity breaking from quantummany-body scars,” Nat. Phys. , 745 (2018).[44] Wen Wei Ho, Soonwon Choi, Hannes Pichler, andMikhail D. Lukin, “Periodic orbits, entanglement, andquantum many-body scars in constrained models: Matrixproduct state approach,” Phys. Rev. Lett. , 040603(2019).[45] Christopher J. Turner, Jean-Yves Desaules, KieranBull, and Zlatko Papi´c, “Correspondence principle formany-body scars in ultracold Rydberg atoms,” (2020),arXiv:2006.13207 [quant-ph].[46] A. A. Michailidis, C. J. Turner, Z. Papi´c, D. A. Abanin,and M. Serbyn, “Stabilizing two-dimensional quantumscars by deformation and synchronization,” Phys. Rev.Research , 022065 (2020).[47] Cheng-Ju Lin, Vladimir Calvera, and Timothy H. Hsieh,“Quantum many-body scar states in two-dimensionalRydberg atom arrays,” Phys. Rev. B , 220304 (2020).[48] Dolev Bluvstein, Ahmed Omran, Harry Levine, Alexan-der Keesling, Giulia Semeghini, Sepehr Ebadi, Tout T.Wang, Alexios A. Michailidis, Nishad Maskara, Wen WeiHo, Soonwon Choi, Maksym Serbyn, Markus Greiner,Vladan Vuletic, and Mikhail D. Lukin, “Controllingmany-body dynamics with driven quantum scars in ryd-berg atom arrays,” (2020), arXiv:2012.12276 [quant-ph].[49] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,and Z. Papi´c, “Quantum scarred eigenstates in a Rydbergatom chain: Entanglement, breakdown of thermalization,and stability to perturbations,” Phys. Rev. B , 155134(2018).[50] Vedika Khemani, Chris R. Laumann, and AnushyaChandran, “Signatures of integrability in the dynamicsof Rydberg-blockaded chains,” Phys. Rev. B , 161101(2019).[51] Cheng-Ju Lin, Anushya Chandran, and Olexei I.Motrunich, “Slow thermalization of exact quantummany-body scar states under perturbations,” Phys. Rev.Research , 033044 (2020).[52] Ian Mondragon-Shem, Maxim G. Vavilov, and IvarMartin, “The fate of quantum many-body scars in thepresence of disorder,” (2020), arXiv:2010.10535 [cond-mat.quant-gas]. [53] Vedika Khemani, Michael Hermele, and Rahul Nandk-ishore, “Localization from hilbert space shattering: Fromtheory to physical realizations,” Phys. Rev. B ,174204 (2020).[54] Shriya Pai and Michael Pretko, “Dynamical scar statesin driven fracton systems,” Phys. Rev. Lett. , 136401(2019).[55] Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore,Nicolas Regnault, and B Andrei Bernevig, “Ther-malization and its absence within Krylov subspacesof a constrained Hamiltonian,” arXiv e-prints (2019),arXiv:1910.14048 [cond-mat.str-el].[56] Pablo Sala, Tibor Rakovszky, Ruben Verresen, MichaelKnap, and Frank Pollmann, “Ergodicity breaking aris-ing from hilbert space fragmentation in dipole-conservinghamiltonians,” Phys. Rev. X , 011047 (2020).[57] Sebastian Scherg, Thomas Kohlert, Pablo Sala, FrankPollmann, Bharath H M, Immanuel Bloch, andMonika Aidelsburger, “Observing non-ergodicity dueto kinetic constraints in tilted Fermi-Hubbard chains,”arXiv:2010.12965v1.[58] M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann,“Stark many-body localization,” Phys. Rev. Lett. ,040606 (2019).[59] Evert van Nieuwenburg, Yuval Baum, and Gil Re-fael, “From bloch oscillations to many-body localizationin clean interacting systems,” PNAS , 9269–9274(2019). [60] Ruixiao Yao and Jakub Zakrzewski, “Many-body lo-calization of bosons in an optical lattice: Dynamicsin disorder-free potentials,” Phys. Rev. B , 104203(2020).[61] J. M. Deutsch, “Quantum statistical mechanics in aclosed system,” Phys. Rev. A , 2046 (1991).[62] Mark Srednicki, “Chaos and quantum thermalization,”Phys. Rev. E , 888 (1994).[63] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss,“Schrieffer–wolff transformation for quantum many-bodysystems,” Ann. Phys. , 2793 – 2826 (2011).[64] Fabian H. L. Essler, Holger Frahm, Frank G¨ohmann,Andreas Kl¨umper, and Vladimir E. Korepin, Theone-dimensional Hubbard model (Cambridge UniversityPress, 2005).[65] “Supplemental online material,”.[66] Vadim Oganesyan and David A. Huse, “Localization ofinteracting fermions at high temperature,” Phys. Rev. B , 155111 (2007).[67] Madan Lal Mehta, Random matrices , Vol. 142 (Elsevier,2004).[68] Igor Lesanovsky and Hosho Katsura, “Interacting Fi-bonacci anyons in a Rydberg gas,” Phys. Rev. A ,041601 (2012).[69] Nicola Pancotti, Giacomo Giudice, J. Ignacio Cirac,Juan P. Garrahan, and Mari Carmen Ba˜nuls, “Quan-tum east model: Localization, nonthermal eigenstates,and slow dynamics,” Phys. Rev. X , 021051 (2020). Supplementary online material for “A proposal for realising quantum scars in thetilted 1D Fermi-Hubbard model”
Jean-Yves Desaules , Ana Hudomal , , Christopher J. Turner , and Zlatko Papi´c School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom Institute of Physics Belgrade, University of Belgrade, 11080 Belgrade, SerbiaIn this Supplementary Material, we discuss details of the numerical implementation of the tilted 1D Fermi-Hubbard model using theJordan-Wigner mapping to spins, the symmetries of the model, and we present more details on the level statistics and eigenstate propertiesof the model. We provide further evidence for the importance of the hypergrid subgraph for many-body scarring, including a perturbationwhich improves the revivals. Finally, we also explain how our results relate to a different regime of the model at high tilt (∆ (cid:29)
U, J ) andelectronic filling ν = 1 / JORDAN-WIGNER TRANSFORMATION ANDSYMMETRIESJordan-Wigner transformation
In order to simplify the computations, we perform aJordan-Wigner transformation to express the Hamilto-nian in terms of spin operators. As a convention, wewill set that the spin-down fermions are located “on theright” of the up-spin ones. For open boundary conditionsthe resulting Hamiltonian for the effective model is :ˆ H eff , JW = − J (cid:88) j ˆ S + j, ↓ ˆ S − j +1 , ↓ ˆ P j, ↑ (cid:16) − ˆ P j +1 , ↑ (cid:17) + J (cid:88) j ˆ S + j, ↑ ˆ S − j +1 , ↑ ˆ P j, ↓ (cid:16) − ˆ P j +1 , ↓ (cid:17) (S1)whereˆ S z = (cid:18) − (cid:19) , ˆ S + = (cid:18) (cid:19) and ˆ S − = (cid:18) (cid:19) are the usual Pauli spin operators and ˆ P j,σ =
1+ ˆ S zj,σ isthe projector on the excited state of the Jordan-Wignerspin. In order to minimise the confusion between the spinof the original fermions and the Jordan-Wigner spins wewill use “spin” to only refer to the fermion spin. For theJordan-Wigner spins we will describe the physics in termof “excitations”, e.g., the action of ˆ S + j, ↓ is to create anexcitation on site j with spin down. Symmetries
In the component of the Hilbert space we are interestedin (i.e., the one containing the state with alternating ↑ and ↓ fermions, as explained in the main text), the Hamil-tonian in the Jordan-Wigner spin representation has two Z symmetries. The first one, denoted as ˆ z , is given bythe joint action of the doublon parity operator and the spin inversion operator:ˆ z = N (cid:89) j =1 ( − ˆ P j, ↓ ˆ P j, ↑ (cid:32) ˆ S + j, ↑ ˆ S − j, ↓ + ˆ S + j, ↓ ˆ S − j, ↑ + 1+ ˆ S zj, ↑ ˆ S zj, ↓ (cid:33) (S2)The doublon parity is diagonal in the product basis andgives +1 if there is an even number of doublons and -1otherwise. The spin inversion simply changes the excita-tions between up spin and down spin.The second symmetry, denoted as ˆ z , corresponds tothe joint application of the spatial inversion operator andof the particle-hole operator:ˆ z = N/ (cid:89) j =1 (cid:89) σ (cid:16) ˆ S + j,σ ˆ S + N − j − ,σ + ˆ S − j,σ ˆ S − N − j − ,σ + 1 − ˆ S zj,σ ˆ S zN − j − ,σ (cid:17) . (S3)The spatial inversion swaps sites j and N − j , whilethe particle-hole conjugation puts an excitation in everyempty site and vice-versa. Performing a Jordan-Wignertransformation on the full Fermi-Hubbard model with thesame fermion ordering convention produces a spin Hamil-tonian which has the symmetry generated by ˆ z but notby ˆ z .Working in the fermionic language, the correspondingset of symmetry operators areˆ y = (cid:89) j e − iπ ( i ˆ c † j, ↓ ˆ c j, ↑ − i ˆ c † j, ↑ ˆ c j, ↓ ) (S4)and ˆ y = (cid:89) j ( − ˆ n j, ↑ ˆ n j, ↓ PR , (S5)using the particle-hole operator P and spatial-reflectionoperator R defined by the adjoint actions, P c † j,σ = c j,σ P , R c † j,σ = c † N − j − ,σ R . (S6)The particle-hole operator also has a non-trivial actionon the vacuum state P| (cid:105) = (cid:89) j c † j, ↑ c † j, ↓ | (cid:105) . (S7)These are equivalent to ˆ z and ˆ z respectively, up to anoverall spin rotation and some charge-dependent phasefactors. The first one is similar to the spin-reversal sym-metry from Eq. (2.57) in Ref. [64]. Negative matrix elements
The spin formulation of the model, after the Jordan-Wigner transformation, offers a convenient framework fornumerics. However, it is clear that the Hamiltonian inEq. (S1), expressed in the product state basis, has bothnegative as well as positive matrix elements. In orderto treat this model as an unweighted graph, we showthat a simple basis transformation is enough to get ridof all minus signs in the Hamiltonian. These occur onlywhen moving a spin down excitation. Here we proposethe following convention for fixing the signs: we assignto each product a sign equal to the parity of the numberof spin down excitations on the even sites. This corre-sponds to a change of basis using the diagonal matrixˆ T = (cid:80) N/ j =1 ( − ˆ P j, ↓ . If a spin up excitation is moved,this does not change this quantity and it follows thatboth states connected by this move have the same sign.Whether it is +1 or -1 is irrelevant as the matrix elementwill get a factor equivalent to their product, which is al-ways +1. On the other hand, if a spin down excitationis moved, this parity number will always change by 1,and the states connected by the move will have oppositesigns. From this, it follows that all the matrix elements ofˆ T ˆ H eff , JW ˆ T are equal to +1, thus we can view the modelin terms of an undirected, unweighted graph.Alternatively, the same result can be obtained bychoosing a different convention for the Jordan-Wignertransformation. Instead of interleaving the fermions witha different spin-projection, one can first place all up-spinfermions and then all down-spin ones with a simple lin-ear order for each species. This has the effect of ensuringthat each of the ‘hopping’ terms never takes a fermionpast another. Then, all these matrix elements are equaland can be brought to 1 by the choice of the couplingconstant J . INDIVIDUAL SYMMETRY SECTORS
Because of SU(2) symmetry, the number of total spinsectors is extensive in system size. In Fig. S1 we showresults for the overlap of eigenstates with |− + (cid:105) state,while Fig. S2 shows the entropy of eigenstates in differenttotal spin sectors. We focus on system size N = 16 andneglect the sector with maximum total spin ( S = 8),since this only has a single state. All sectors displaytowers of states with the same approximate spacing ofenergy equal to 2 √ N/ Figure S1. Overlap of the eigenstates with |− + (cid:105) state in theeffective model for N = 16 in each total spin sector. Thered crosses indicate the states at the top of each tower in therelevant sector. − S . Because of the change in parity of that quantity,the location in energy of the towers is also dependent onthe parity of the total spin. Sectors with an odd numberof towers will have them at E ≈ ± n √
2, with n beinga positive integer or zero. On the other hand, sectorswith an even number of towers will have them at E ≈± (1 + 2 n ) √ S ≤
4) the entropy band isrelatively narrow, with only a few outliers correspondingto the top state of some of the towers (Figure S2). In-terestingly, the states with a very low entropy are onlypresent for odd total spin for N = 16. This is linkedwith the dependence of the location on the towers on thetotal spin. Indeed, the most atypical states have a fixedenergy E ≈ ±√ N = 16.After all the symmetries have been resolved, one canstudy energy level statistics in each symmetry sector sep-arately (Figure S3). All sectors with a large number ofstates ( D > ) have mean level spacing (cid:104) r (cid:105) very closeto 0.53, compatible with the Wigner-Dyson distribution.We also show a histogram of level spacing after unfolding0 Figure S2. Entropy of the eigenstates with |− + (cid:105) state in theeffective model for N = 16. The red crosses indicate thestates at the top of each tower in the relevant sector and arethe same as in Figure S1. [67]. This is done for the sector z = 1, z = 1, S = 2in Figure S4. This sector was chosen because it has thelargest overlap with |− + (cid:105) state at the system size studiedand also has a large number of states. The histogram ofthe level spacing confirms that this sector indeed corre-sponds to a chaotic quantum system. INFLUENCE OF THE HYPERGRID
The influence of the hypergrid subgraph is visible inthe dynamics, i.e., in the revivals from a few productstates as demonstrated in the main text. In this sectionwe illustrate the impact of the hypergrid on eigenstatesand the long-time behaviour of the system.In a fully ergodic system, one expects the support ofan eigenstate on a subset of states to be approximatelyequal to the ratio of the number of states belonging tothis subset and the total dimension of the Hilbert space.Performing this computation for the effective model andchoosing the hypergrid subgraph as the subset showsanomalous concentration of some eigenstates – see Fig-ure S5. As the overlap with the |− + (cid:105) state forms a lower
1, 1 1, -1 -1, 1 -1, -1Symmetries T o t a l s p i n ( . . . h r i Figure S3. Number of states and level statistics in each sym-metry sector ( z , z
2) in the effective model for N = 16. bound to the support on the hypergrid, we find the samekind of towers of states, located at the same energies,in this plot. However, where the maximum overlap with |− + (cid:105) was approximately 0.011 for N = 16, the maximumsupport on the hypergrid is instead close to 0.76 for thesame size. This confirms that the hypergrid subgraphstill leaves a large imprint on the eigenstates, even whenit only comprises less than 2 .
5% of the Hilbert space for N = 16.Another signature of non-ergodicity caused by the hy-pergrid subgraph can be seen in the long-time expecta-tion value of the projector onto the hypergrid, P HG ( t ),when starting from the |− + (cid:105) state. It converges to a non- s . . . . . . P r o b a b ili t y d e n s i t y PoissonWDSemi-Poisson h r i = 0 . Figure S4. Distribution of the energy level spacing after un-folding [67] in the sector z = 1, z = 1, S = 2 for N = 16.The result for this sector is very close to a Wigner-Dysondistribution. Figure S5. Expectation value of the projector onto the hy-pergrid for the eigenstates at N = 16. The colours indicatethe value of total spin for each eigenstate and the dashed lineis the average value of P HG for the system. Crosses are thosestates with highest overlap around their energy irrespectiveof spin. zero value which is higher than what would be expectedfrom the relative size of the hypergrid in the Hilbert space(Figures S6 and S7). This hints that the embedding ofthe hypergrid in the effective model is non-trivial andthat the long-lived oscillations in the dynamics cannotsimply be explained by the relative size of this substruc-ture. t . . . . . . P H G ( t ) N =6 N =8 N =10 N =12 N =14 N =16 N =18 N =20 Figure S6. Expectation value of the projector on the hyper-grid states over time for various system sizes with the initialstate |− + (cid:105) . As the system size increases, the time necessaryto leak out of the hypergrid gets longer. FULL MODEL
In this section we provide further results showing thatthe scarred dynamics persists in the full model in Eq. (1)of the main text at finite values of ∆ and U . In our anal-ysis of the effective model, we looked at the fidelity ofwavefunction revivals; here we show that the wavefunc-tion revivals are also visible in the full model for a rela-tively broad range of parameters U and ∆ (Figure S8). N − − l og ( P H G ( t = ∞ )) D HG / D , fit slope -0.132Dynamics, fit slope -0.081 Figure S7. Expectation value of the projector on the hyper-grid states after a long time (starting from the |− + (cid:105) state)compared to the expected value from the relative size of thehypergrid. The long-time value of P HG ( t ) is computed as theaverage between t = 80 and t = 100 of the data in Figure S6.The linear fits are taken for N = 14 to 20. While the scaling isindeed exponential, the exponent is smaller than what wouldbe expected from the size of the Hilbert space. We also plot the overlap of |− + (cid:105) state with the eigen-states of the full model in Figure S9. Colouring eacheigenstate with the expectation value of U (cid:80) j (cid:104) ˆ n j, ↑ ˆ n j, ↓ (cid:105) +∆ (cid:80) j,σ j (cid:104) ˆ n j,σ (cid:105) highlights the fact that the eigenstatesstart to form distinct towers at these parameter values.Each of these towers correspond to an expectation valueof this operator with a value close to an integer multipliedby U = ∆. As this parameter is increased the mixing be-tween the towers is reduced until they are completelyseparated in the U = ∆ = ∞ limit. In this case onlyeigenstates in the central tower have a non-zero overlapwith the |− + (cid:105) state. U ∆ . . . f Figure S8. Maximum revival fidelity in the full model between t = 1 and t = 10 when starting from the |− + (cid:105) state. Therevivals show an increasingly better fidelity on the U = ∆diagonal as the value of these parameters get larger. Thehigh fidelity in the top left corner corresponds to the high-tiltregime investigated in [57], where a large number of statesshow revivals due to the conservation of the dipole momentand of the number of doublons separately . Figure S9. Overlap of |− + (cid:105) state with eigenstates of the fullmodel for N = 10, U = ∆ = 5, and J = 1. The colourindicates the expectation value of the number of doublonsplus the dipole moment. We see that already at this value ofthe parameters the eigenstates start to form distinct towers. IMPROVING THE REVIVAL BYPERTURBATION
As the revivals emerge due to an embedded subgraph,we can devise a perturbation that enhances the revivalsby decreasing the leakage out of the hypergrid. For theeffective model, this can be achieved by a dimerised per-turbation given by H Pert , eff = λJ N/ − (cid:88) j =1 (cid:88) σ ˆ c † j,σ ˆ c j +1 ,σ ˆ n j,σ (1 − ˆ n j +1 ,σ )+h . c ., (S8)i.e., the perturbation only affects interaction between theunit cells, leaving the hypergrid subgraph untouched. For λ = 1, the hopping term between cells in the Hamiltonianand in the perturbation cancel each other. As a result,the hypergrid becomes completely isolated from the restof the Hilbert space, resulting in the perfect wavefunctionrevival. This can be seen in Figure S10 for N = 14. Theseresults show clearly that the perturbation has a very weakeffect on the period of the revivals, but only modulatestheir amplitude. Furthermore, as expected the fidelitypeaks at λ = 1, when the system revives perfectly.The same perturbation can also be applied to the fullmodel, in which case it takes the formˆ H = λJ N/ − (cid:88) j =1 (cid:88) σ ˆ c † j,σ ˆ c j +1 ,σ + h . c .. (S9)The optimal value of λ for the first peak of the fidelityrevivals is also present at λ = 1, but the revivals are nolonger perfect (the first peak reaches ∼ t . . . . . . | h ψ ( ) | ψ ( t ) i | − . . . . . . . . λ λ . . . f Figure S10. Fidelity revivals for the perturbed effective modelfor N = 14. At λ = 1 the hypergrid subgraph is completelyisolated from the rest of the Hilbert space and the revivalsfrom the |− + (cid:105) state (and from any other of the hypergridcorners) become perfect. that the oscillatory is also due to the imprint of the hy-pergrid in this case. FILLING ν = 1 / While most of our work has focused on the sector withfilling ν = 1, we find similar dynamics in the ν = 1 / (cid:29) U, J , the effective Hamiltonian is given byˆ H dipeff = J (3) ˆ T + U (cid:16) − J ∆ (cid:17) (cid:88) i ˆ n j, ↑ ˆ n j, ↓ + 2 J (3) ˆ T XY + 2 J (3) (cid:88) j,σ ˆ n j,σ ˆ n j +1 , ¯ σ , (S10) t . . . . . . | h ψ ( ) | ψ ( t ) i | − . . . . . . . . λ λ . . . f Figure S11. Fidelity revivals for the full model with pertur-bation for N = 12 starting from the |− + (cid:105) state. As for theeffective model λ = 1 gives the best revivals for the this state. J (3) = J U ∆ (S11)ˆ T = (cid:88) i,σ ˆ c i,σ ˆ c † i +1 ,σ ˆ c † i +1 , ¯ σ ˆ c i +2 , ¯ σ + h.c. , (S12)ˆ T XY = (cid:88) i,σ ˆ c † i, ¯ σ ˆ c i +1 , ¯ σ ˆ c † i +1 ,σ ˆ c i,σ (S13)The action of this Hamiltonian also produces a fragmen-tation of the Hilbert space, and we focus on the sectorwith N/ N/ T operator only al-lows to squeeze two fermions with opposite spin to createa doublon between them. As all fermions are originallyseparated by an empty site it is impossible for this op-erator to reach a state with neighbouring fermions: theywill always be either on the same site or with an emptysite between them. Hence, the operator ˆ T XY and the po-tential terms ˆ n j,σ ˆ n j +1 , ¯ σ are always zero. As the doublon-counting term is diagonal, the only term that allows forhopping between product states is ˆ T . By studying theeffect of this operator in a 4-site cell we will show that itleads to the same constrained Hilbert space structure asthe sector studied at filling ν = 1. Indeed, for ν = 1 with two sites we have the possiblestates | ↑↓(cid:105) ↔ | (cid:108) (cid:105) ↔ | ↓↑(cid:105) forming a three level system.For ν = 1 /
2, the operator ˆ T also only allows 3 states butnow for four sites as | ↑ ↓ (cid:105) ↔ | (cid:108) (cid:105) ↔ | ↓ ↑ (cid:105) . Asall actions of the Hamiltonian can be understood in termsof these cells, it ensues that both models have exactly thesame off-diagonal matrix elements if N ν =1 / = 2 N ν =1 .The only difference is the diagonal term counting thenumber of doublons for ν = 1 / H CDW = J (3) ˆ T + U (cid:16) − J ∆ (cid:17) (cid:88) i ˆ n j, ↑ ˆ n j, ↓ = J (3) (cid:34) ˆ T + (cid:16) ∆ J − (cid:17) (cid:88) i ˆ n j, ↑ ˆ n j, ↓ (cid:35) . (S14)As the Hamiltonian in Eq. (S10) is only valid for ∆ (cid:29) U, J , this is also the case for the one in Eq. (S14). How-ever, it is easy to see that as ∆ /J increases, the potentialterm will dominate the dynamics. So there is no way toobtain the Hamiltonian ˆ T on its own. Set ∆ too lowand the higher order terms in the Schrieffer-Wolff trans-formation will not be negligible, set it too high and thepotential term will dominate the dynamics. On the otherhand, this issue does not arise in the ν = 1 sector as onlythe equivalent of ˆ T3